Question

1. the side views of two different ramps are shown below. which ramp shows a greater slope? explain.

Explanation:

Step1: Recall slope formula for ramp

The slope of a ramp (a right triangle's hypotenuse slope) is calculated as \( \text{slope} = \frac{\text{rise}}{\text{run}} \), where rise is the vertical height and run is the horizontal length.

Step2: Calculate slope of Ramp A

For Ramp A, rise = 20 feet, run = 30 feet.
Slope of A: \( \frac{20}{30} = \frac{2}{3} \approx 0.6667 \).

Step3: Calculate slope of Ramp B

For Ramp B, rise = 14 feet, run = 21 feet.
Slope of B: \( \frac{14}{21} = \frac{2}{3} \)? Wait, no: \( \frac{14}{21} = \frac{2}{3} \)? Wait, 14÷7=2, 21÷7=3, so \( \frac{14}{21} = \frac{2}{3} \)? Wait, no, 14/21 simplifies to 2/3? Wait, 20/30 is 2/3, 14/21 is also 2/3? Wait, that can't be. Wait, 20 divided by 30 is 2/3 ≈0.6667. 14 divided by 21: 14÷21 = 2÷3 ≈0.6667. Wait, but maybe I miscalculated. Wait, 14/21: divide numerator and denominator by 7: 2/3. 20/30: divide by 10: 2/3. Wait, so they have the same slope? But that seems odd. Wait, no, wait the problem: maybe the run or rise is different. Wait, Ramp A: rise 20, run 30. Ramp B: rise 14, run 21. So 20/30 = 2/3, 14/21 = 2/3. So both slopes are 2/3. But maybe I made a mistake. Wait, let me check again. 20 divided by 30: 20/30 = 2/3 ≈0.6667. 14/21 = 2/3 ≈0.6667. So they have the same slope. But the question is "which ramp shows a greater slope". Wait, maybe I misread the numbers. Wait, Ramp A: rise 20, run 30. Ramp B: rise 14, run 21. So 20/30 = 2/3, 14/21 = 2/3. So their slopes are equal. But maybe the problem has a typo, or I misread. Wait, no, 20/30 is 2/3, 14/21 is 2/3. So both slopes are the same. But if we calculate decimal: 20÷30≈0.6667, 14÷21≈0.6667. So they are equal. But the question says "which ramp shows a greater slope". Maybe I made a mistake. Wait, let's re-express: 20/30 = 0.666..., 14/21 = 0.666.... So they are equal. But perhaps the problem intended different numbers. Wait, maybe the run for Ramp B is 14 and rise 21? No, the diagram shows Ramp B: vertical side 14 (rise), horizontal side 21 (run). So according to calculation, both slopes are 2/3. But maybe the problem is designed to show that. Alternatively, maybe I miscalculated. Wait, 20 divided by 30: 20/30 = 2/3 ≈0.6667. 14 divided by 21: 14/21 = 2/3 ≈0.6667. So they have the same slope. But the question asks "which ramp shows a greater slope". Maybe the problem has a typo, or I misread the numbers. Wait, maybe the run for Ramp A is 20 and rise 30? No, the diagram: Ramp A has vertical side 20 (rise), horizontal side 30 (run). Ramp B: vertical side 14 (rise), horizontal side 21 (run). So slope is rise over run. So both are 2/3. So they have the same slope. But perhaps the problem intended different values. Wait, maybe I made a mistake in the slope formula. Slope is rise over run, yes. So for a ramp, slope = height (rise) / length (run). So yes, that's correct. So both ramps have the same slope. But the question says "which ramp shows a greater slope". Maybe the problem has an error, or I misread. Wait, let me check again. 20/30 = 2/3 ≈0.6667. 14/21 = 2/3 ≈0.6667. So they are equal. So the answer would be that both ramps have the same slope, because their rise over run ratios are equal (20/30 = 14/21 = 2/3).

Answer:

Both Ramp A and Ramp B have the same slope, because the slope (rise/run) for Ramp A is \( \frac{20}{30} = \frac{2}{3} \) and for Ramp B is \( \frac{14}{21} = \frac{2}{3} \), so their slopes are equal.

(Note: If there was a miscalculation, recheck the problem's numbers. But based on given values, slopes are equal.)